The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 45 



K = a cos (^ — ?'')+ T j~ s"^ (^ ~ '')• 







These two sets of equations correspond to (5) and (6) of the first section 



_, r- da . . , ^s da . . . ^ , ^ da 



hrom -^ -J- = sin I cos (« — j), 7^ = — sm i cos (v — 3), and -p — coi 



sin (v — 3); we easily find 



da _ r- da 

 db~~ Ti di 

 dK _ p- dx 



da 



di ' 



dK 



cos I 

 sin V 

 cos i 



-J-. = K —. :, 



di sm I 



(3) 



wliich may be found useful. 



The part of the disturbance function depending on the inclination of the 

 orljit to the fixed plane is very troublesome to express by means of i, 3, and 0, 

 and the corresponding quantities i', 3', and 6', relative to the disturbing body ; 

 and when either 3- is expressed by means of 6, 3' by means of O", or the latter 

 by means of the former, would contain a great number of terms. But it may 

 be expressed very simply without these quantities, by means of the latitude (p 

 and the reduction A, and the corresponding quantities 0' and A' relative to 

 the disturbing body. Thus we should have u — A, and v' — A', and v' — A' for 

 the longitudes on the fixed plane, and 



Jj "U z 



- = cos COS (v — A), -_ = cos sin (?; — A), - = sin , 



x' y' z' 



— = COS 0' COS {v' - A'), ~ = cos 0' sin {v' — A'), — = sin 0'. 



T3 . xx' + yy' + zz' , „ 

 But cos Y = ^^^ ; therefore, 



cos x = cos cos 0' ]cos (y- A) cos («'— A') + sin (u — A) sin (w' - A') ( + 

 sin sin 0' = cos cos 0' cos {v — v' — A + A') + sin sin 0' = {l-a-f 



{l-a'-)Hcos{v-v')+sm{v-v'){A-A'ycos{v-v')^^—^--&.c.\->raa'. (4) 



